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<title>Simplified HKF model</title>
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<p>The <b>simplified Helgeson-Kirkham-Flowers (HKF) model:</b>
<table cellspacing="0" cellpadding="0" border="0">
<tr valign="top"><td><tt>&nbsp;&nbsp;&nbsp;</tt></td>
	<td valign="top"><nobr>log <i>&#947;<sub>i</sub></i><font size="+2"> </font>= </nobr></td>
	<td valign="top"><nobr>&#8722;<i>A</i><font size="+2"> </font><i>Z<sub>i</sub></i><sup>2</sup></nobr>
		<nobr><font size="+2">(</font>&#8730;<b><i>I</i></b> &nbsp;/
			<font size="+1">&nbsp;(</font>1+<i>B</i> &#8730;<b><i>I</i></b><font size="-2"> </font>
				<font size="+1">)</font><font size="+2">)</font></nobr>
	<nobr>&#8722; log<font size="+1">(</font>1+0.018 &#8721;<i>m<sub>k</sub></i><font size="+1">)</font></nobr>
	<nobr>+ <i>b <b>I</b></i></nobr>
</td></tr></table>
</p>

<p class="cont">
where <b><i>I</i></b> is the <a href="SP_Ionic_Strength.htm">ionic strength</a>,
<nobr>&#947;<sub><i>i</i></sub></nobr> is the activity coefficient of an
ion &#147;<i>i</i>&#148; with electric charge <nobr><i>Z<sub>i</sub></i>;&nbsp;</nobr>
<i>A</i> is the <a href="SP_Ionic_Strength_DH.htm">Debye-H&uuml;ckel</a>
slope (see the table below);&nbsp;
<i>B</i> (= <nobr><i>&aring;B<sub>&#947;</sub></i></nobr> in the
<a href="SP_Ionic_Strength_DH.htm">Debye-H&uuml;ckel <nobr>eqn.</a>)</nobr>
includes the distance of closest approach (effective ionic radii);&nbsp;
and <i>b</i> is a solute specific parameter. The sum
is made over all solute species &#147;<i>k</i>&#148; in the aqeuous solution.
See the <a href="#Refs">list of references</a>.
</p>

<p>
In <a href="S_0_Main.htm">SPANA</a> (and <nobr><a href="SP_0_Main.htm">SED/PREDOM</a>)</nobr>
the HKF model is <b>simplified</b> by using the <b>NaCl</b> parameters:
<ul><li>
the <i>B</i> parameter
<nobr>(= <i>&aring;B<sub>&#947;</sub></i>),</nobr>
which includes the effective ionic radii <nobr>&#147;<i>&aring;</i>&#148;,</nobr>
 is obtained by multyplying
the Debye-H&uuml;ckel constant  <nobr><i>B<sub>&#947;</sub></i></nobr> (from
Table&nbsp;1, <nobr>p.1295,</nobr> in <a href="#Refs">Helgeson <i>et al.</i> <nobr>1981</a>)</nobr>
with the effective ionic distance for NaCl
<nobr>(<i>&aring;</i></nobr> = 1.91+1.81 = <nobr>3.72;&nbsp;</nobr>
 see Tables 2 or 3 in <a href="#Refs">Helgeson <i>et al.</i> <nobr>1981</a></nobr>
 or Table&nbsp;9 in <a href="#Refs">Shock <i>et al.</i> <nobr>1992</a>).</nobr>
 This results in <i>B</i> = 1.22 at <nobr>25&deg;C</nobr> (see table below).</li>
<li>the <i>b</i> parameter for NaCl solutions = 0.064 at
<nobr>25&deg;C</nobr>, see table below (from Table&nbsp;5, <nobr>p.1327,</nobr> in
<a href="#Refs">Helgeson <i>et al.</i> <nobr>1981</a>,</nobr>
and <nobr>Table&nbsp;A-2</nobr> in
<a href="#Refs">Oelkers and Helgeson <nobr>1990</a>).</nobr></li>
<li>for the sum <nobr><font size="+1">(</font>&#8721;<i>m<sub>k</sub></i><font size="+1">)</font>:&nbsp;</nobr>
if a constant value for the ionic strength is given, then the value of <b><i>I</i></b>
is used instead of <nobr>&#8721;<i>m<sub>k</sub></i>.</nobr> However, if the ionic strength is calculated
iteratively at each point, then <nobr>&#8721;<i>m<sub>k</sub></i></nobr> is calculated as well.
</li></ul>
</p>


<p>
The HKF parameters <i>A, B,</i>  and <i>b</i>
for this simplified model are (<a href="#Refs">Helgeson <i>et al.</i> <nobr>1981</a>,</nobr>
<a href="#Refs">Oelkers and Helgeson <nobr>1990</a>):</nobr></p>

<p><center><table cellspacing="0" cellpadding="5" border="1">
<tr><td><nobr><i>t</i> (&deg;C)</nobr></td>
	<td align="center"><i>A</i></td>
	<td align="center"><i>B</i></td>
	<td align="center"><i>b</i></td></tr>
	<tr><td align="right">0</td><td>&nbsp;0.491</td><td>1.21</td><td>0.041</td></tr>
	<tr><td align="right">25</td><td>&nbsp;0.509</td><td>1.22</td><td>0.064</td></tr>
	<tr><td align="right">50</td><td>&nbsp;0.534</td><td>1.24</td><td>0.074</td></tr>
	<tr><td align="right">100</td><td>&nbsp;0.600</td><td>1.27</td><td>0.076</td></tr>
  </table></center></p>

<p>The activity of water <nobr>(H<sub>2</sub>O)</nobr> is given by
<table cellspacing="0" cellpadding="0" border="0">
<tr valign="top"><td><tt>&nbsp;&nbsp;&nbsp;</tt></td>
	<td valign="top"><nobr>log <i>a</i><sub>H<sub>2</sub>O</sub> =<font size="+1"> </font></nobr></td>
	<td valign="top"><nobr>&#8722;<font size="+1"> </font>&Phi; &#8721;<i>m<sub>k</sub></i>&nbsp;/</nobr>
		<nobr><font size="+1">(</font>ln(10)&nbsp;55.508<font size="+1">)</font></nobr>
</td></tr></table></p>

<p class="cont">
where &#147;&Phi;&#148;
is the osmotic coefficient, calculated using the <b>simplified HKF model</b> as follows:
<table cellspacing="0" cellpadding="0" border="0">
<tr valign="top"><td><tt>&nbsp;&nbsp;&nbsp;</tt></td>
	<td valign="top"><nobr>&Phi; =<font size="+1"> </font></nobr></td>
	<td valign="top"><nobr>ln<font size="+1">(</font>1+0.018 &#8721;<i>m<sub>k</sub></i><font size="+1">)/(</font>0.018 &#8721;<i>m<sub>k</sub></i><font size="+1">)</font></nobr>
		<nobr>&#8722; (2/3) <font size="+1">(</font>ln(10)/&#8721;<i>m<sub>k</sub></i><font size="+1">)</font> <i>A</i>&nbsp;<b><i>I</i></b><sup>3/2</sup></nobr>
		<nobr><i>&sigma;</i>(<i>B</i>&#8730;<b><i>I</i></b>)</nobr>
          <nobr>+&nbsp;<font size="+1">(</font>ln(10)/&#8721;<i>m<sub>k</sub></i><font size="+1">)</font>
					(<i>b</i>/2) <b><i>I</i></b></nobr>
</td></tr></table></p>
<p class="cont">where
<table cellspacing="0" cellpadding="0" border="0">
<tr valign="top"><td><tt>&nbsp;&nbsp;&nbsp;</tt></td>
	<td valign="top"><nobr>&sigma;</i>(<i>x</i>)<font size="+2"> </font>= </nobr></td>
	<td valign="top"><nobr>(3/<i>x</i><sup>3</sup>)</nobr> <nobr><font size="+2">(</font>(1+<i>x</i>)</nobr> <nobr>&#8722; 1/(1+<i>x</i>)</nobr> <nobr>&#8722; 2 ln(1+<i>x</i>)<font size="+2">)</font></nobr>
</td></tr></table></p>

<a name="Refs"></a>
<h4>References</h4>
<p>
<ul>
<li>H.C.Helgeson, D.H.Kirkham and G.C.Flowers, <i>Amer. Jour. <nobr>Sci.,</nobr></i> <b>281</b> (1981)
1249-1516; eqns. 165-167, 121, 297, 298</li>
<li>E.H.Oelkers and H.C.Helgeson, <i>Geochim. Cosmochim. Acta,</i> <b>54</b> (1990)
727-738; eqns.22 and 23.</li>
<li>E.L. Shock, E.H.Oelkers, J.W. Johnson, D.A. Sverjensky, H.C.Helgeson
<i>J. Chem. Soc. Faraday Trans.,</i> <b>88</b> (1992)
803-826
</li>
</ul>
</p>

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